shiecldk said:Let A be a 4×3 matrix and let
c=2a1+a2+a3
(a) If N(A) = {0}, what can you conclude about the solutions to the linear system Ax=c?
(b) If N(A) ≠ {0}, how many solutions will the system Ax=c have? Explain.
The null space of a matrix is the set of all vectors that, when multiplied by the matrix, result in a zero vector. In other words, it is the set of vectors that do not change when multiplied by the matrix.
To calculate the null space of a matrix, we need to first reduce the matrix to its reduced row-echelon form using row operations. The columns corresponding to the leading 1's in the reduced row-echelon form matrix will form the basis for the null space.
The dimension of the null space of a matrix is also known as the nullity of the matrix. It is equal to the number of free variables in the reduced row-echelon form of the matrix. In other words, it is the number of columns in the matrix that do not have a leading 1.
The null space of a matrix helps us understand the linear dependence or independence of its columns. If the null space is non-empty, it means that there are linearly dependent columns in the matrix. If the null space is empty, it means that the columns are linearly independent.
The null space of a matrix is used in various applications, such as finding solutions to systems of linear equations, solving differential equations, and in data compression and image processing. It also has applications in fields like physics, engineering, and computer science.